On the Direct Detectability of the Cosmic Dark Ages: 21-cm Emission from Minihalos

Transcription

1 On the Direct Detectability of the Cosmic Dark Ages: 21-cm Emission from Minihalos Ilian T. Iliev 1, Paul R. Shapiro 2, Andrea Ferrara 1, and Hugo Martel 2 iliev@arcetri.astro.it, shapiro@astro.as.utexas.edu arxiv:astro-ph/ v3 10 May 2002 ferrara@arcetri.astro.it, hugo@simplicio.as.utexas.edu ABSTRACT In the standard Cold Dark Matter (CDM) theory of structure formation, virialized minihalos (with T vir 10 4 K) form in abundance at high redshift (z > 6), during the cosmic dark ages. The hydrogen in these minihalos, the first nonlinear baryonic structures to form in the universe, is mostly neutral and sufficiently hot and dense to emit strongly at the 21-cm line. We calculate the emission from individual minihalos and the radiation background contributed by their combined effect. Minihalos create a 21-cm forest of emission lines. We predict that the angular fluctuations in this 21-cm background should be detectable with the planned LOFAR and SKA radio arrays, thus providing a direct probe of structure formation during the dark ages. Such a detection will serve to confirm the basic CDM paradigm while constraining the shape of the power-spectrum of primordial density fluctuations down to much smaller scales than have previously been constrained, the onset and duration of the reionization epoch, and the conditions which led to the first stars and quasars. We present results here for the currently-favored, flat ΛCDM model, for different tilts of the primordial power spectrum. Subject headings: cosmology: theory diffuse radiation intergalactic medium large-scale structure of universe galaxies: formation radio lines: galaxies 1. Introduction No direct observation of the universe during the period between the recombination epoch at redshift z 10 3 and the reionization epoch at z 6 has yet been reported. 1 Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, Firenze, Italy 2 Department of Astronomy, University of Texas, Austin, TX 78712

2 2 While a number of suggestions for the future detection of the reionization epoch, itself, have been made, this period prior to the formation of the first stars and quasars the cosmic dark ages (e.g. Rees 1999) has been more elusive. Standard Big Bang cosmology in the CDM model predicts that nonlinear baryonic structure first emerges during this period, with virialized halos of dark and baryonic matter which span a range of masses from less than 10 4 M to about 10 8 M which are filled with neutral hydrogen atoms. The atomic density n H and kinetic temperature T K of this gas are high enough that collisions populate the hyperfine levels of the ground state of these atoms in a ratio close to that of their statistical weights (3:1), with a spin temperature T S that greatly exceeds the excitation temperature T = K. Since, asweshallshow, T S > T CMB, thetemperatureofthecosmicmicrowave Background (CMB), as well, for the majority of the halos, these minihalos can be a detectable source of redshifted 21-cm line emission. The direct detection of minihalos at such high redshift would be an unprecedented measure of the density fluctuations in the baryons and of the total matter power spectrum at small scales, which will not be probed by other methods yet discussed (e.g. CMB anisotropy). The possibility of 21-cm line emission or absorption by neutral H at high redshift has been considered before (Hogan & Rees 1979; Scott & Rees 1990; Subramanian & Padmanabhan 1993; Kumar, Padmanabhan, & Subramanian 1995; Bagla, Nath, & Padmanabhan 1997; Madau, Meiksen & Rees 1997; Shaver, et al. 1999; Tozzi et al. 2000). Prior to the release of radiation by nonlinear baryonic structures which condense out of the background universe, the spin temperature of H I in the diffuse, uncollapsed gas in the intergalactic medium (IGM) is coupled to the CMB, so that T S = T CMB and neither emission nor absorption in the 21-cm line is possible. Recently, attention has focused on the possibility that radiation from early stars and quasars might decouple T S from T CMB by Lyα pumping resonant scattering in the H Lyα transition followed by decay of the upper state n = 2 to the ground state n = 1 into one or the other of the hyperfine levels (Madau et al. 1997; Tozzi et al. 2000). This mechanism, it has been suggested, will operate on HI in the diffuse, uncollapsed IGM during reionization, first to make T S < T CMB, so that the 21-cm transition can be seen in absorption against the CMB, until the same Lyα scattering heats the gas shortly thereafter and makes T S > T CMB, thereby causing 21-cm emission in excess of the CMB, before reionization finally destroys the H I. In what follows, however, we show that a substantial fraction of the baryons in the universe may already have condensed out of the diffuse IGM into virialized minihalos, prior to and during reionization. Under these conditions, collisional excitation alone is sufficient to decouple T S from T CMB and cause 21-cm emission in excess of the CMB, thereby providing a signature of the cosmic dark ages and of their retreat during reionization.

3 cm Emission from Individual Minihalos The 21-cm emission from a single halo depends upon its internal atomic density, temperature, and velocity structure. We model each CDM minihalo here as a nonsingular, truncated isothermal sphere ( TIS ) of dark matter and baryons in virial and hydrostatic equilibrium, in good agreement with the results of gas and N-body simulations from realistic initial conditions (Shapiro, Iliev & Raga 1999; Iliev & Shapiro 2001, 2002). This model uniquely specifies the internal structure of each halo (e.g. total and core sizes, r t and r 0, central total mass density ρ 0, dark matter velocity dispersion σ V = (4πρ 0 r 2 0 )1/2, and gas temperature T K = µm p σ V /k B, where µ is the mean molecular weight), for a given background cosmology as functions of two parameters, the total mass M and collapse redshift z coll. The minihalos which contribute significantly to the 21-cm emission span a mass range from M min to M max which varies with redshift. M min is close to the Jeans mass of the uncollapsed IGM prior to reionization, M J = (Ω 0 h 2 /0.15) 1/2 (Ω b h 2 /0.02) 3/5 [(1+z)/10] 3/2 M, while M max = (Ω 0 h 2 /0.15) 1/2 [(1+z)/10] 3/2 is the mass for which T vir = 10 4 K according to the TIS model (Iliev & Shapiro 2001) (since halos with T vir 10 4 K are largely collisionally ionized). Halos with T vir 10 4 K may have radiatively cooled gas inside them which would add to the signal we compute, but such gas is expected to lead to the formation of internal sources of ionizing radiation which will largely offset the effect. Since these additional effects are highly uncertain and are related to the onset of radiative feedback and reionization which we are neglecting in these calculations, we will not consider the role of higher temperature halos further. The flux per unit frequency, F ν (df/dν) rec, received at redshift z = 0 at frequency ν rec from a minihalo at redshift z which emits at frequency ν em = ν rec (1+z) is expressed in terms of the brightness temperature T b,em = T b,rec (1+z) according to F νrec = 2ν2 rec c 2 k BT b,rec ( Ω) halo, (1) where ( Ω) halo = πr 2 t /D2 A = π( θ halo/2) 2 is the solid angle subtended by the minihalo, and D A is the angular diameter distance. The brightness temperature T b,em is determined by solving the equation of radiative transfer to derive the brightness profile of the minihalo and integrating this profile over the projected surface area, as follows. The brightness temperature along a line of sight thru a minihalo at projected distance r from the center obeys the equation T b (r) = T CMB e τ(r) + τ(r) 0 T S e τ dτ, (2)

4 4 where quantities are defined in the comoving frame of the minihalo, frequency ν refers here and henceforth to ν em, τ(r) is the total optical depth thru the halo, and the effective absorption coefficient κ ν is given when T T S by: κ ν = 3c2 A 10 n HI 32πν 2 f(ν) T T S (3) (Field 1958), where A 10 = s 1 is the Einstein A-coefficient for the 21-cm transition and f(ν) is the normalized line profile. The spin temperature T S is determined by the balance between collisional and radiative excitation and de-excitation by atoms and electrons and by CMB and Lyα photons, respectively, according to T S = T CMB +y c T K +y α T α 1+y c +y α, (4) where T α is the color temperature of the Lyα photons, and y α and y c are radiative and collisional excitation efficiencies, respectively (Purcell & Field 1956; Field 1958, 1959). The efficiency y c includes contributions from H 0 H 0 collisions, y H, and from e H 0 collisions, y e. Prior to the reionization epoch, Lyα pumping is unimportant and collisional excitation alone must compete with excitation by the CMB. This is only possible for gas that is highly nonlinear and sufficiently hot. Such conditions are achieved only inside virialized halos. As shown in Figure 1, the optical depth of an individual halo is not negligible, particularly for smaller-mass halos (due to their lower T S ). Since T S varies with radial position inside the halo, as a result of its significant central concentration, we must integrate equation (2) numerically. The face-averaged T b of this single halo is given by T b halo ( T b (r)da)/a, where A(M,z) is the geometric cross-section of a halo of mass M and collapse redshift z. The observed flux from an individual halo is then expressed with respect to the CMB by the differential antenna temperature δt b [ T b halo T CMB (z)]/(1+z). The line-integrated flux F(M,z) received from this minihalo is equal to the flux calculated for ν = ν 0 multiplied by a redshifted effective line-width ν eff (z), defined by ν eff (z) ( Fdν)/F ν0. For an optically thin minihalo, ν eff reduces to ν eff (z) = [f(ν 0 )(1+z)] 1. In that case, for a thermal-doppler-broadened line profile, ν eff (z) = [(2πµ) 1/2 ν 0 σ V /c](1+z) 1. We have checked that this approximation is adequate even for the optically thicker halos at the small-mass end of the mass function. The differential line-integrated flux δf(m, z) is given by replacing T b,rec in equation (1) by δt b and integrating over frequency as described above. Our results for individual minihalos are summarized in Figure 1. Line profiles of different minihalos along the same line of sight should not typically overlap. The proper mean free path λ mfp = n halo σ halo 1 for photons to encounter minihalos in ΛCDM is 160 kpc at z = 9

5 5 (Shapiro 2001), corresponding to a frequency separation, ν sep ν 0 H(z)λ mfp /[c(1+z)] 0.1MHz ν eff 10kHz. These results predict a 21-cm forest of minihalo emission lines. At z = 9, for example, there are about 160 minihalo lines per unit redshift along a typical line of sight in an untilted ΛCDM universe (Shapiro 2001). Detecting the stronger lines would require sub-arcsecond spatial resolution, 1 khz frequency resolution, and njy sensitivity. SKA is expected to have sufficient resolution for such observation, but probably not sufficient sensitivity cm Radiation Background from Minihalos The average differential flux per unit frequency relative to that of the CMB from all the minihalos observed within a given beam of angular size θ beam and frequency bin ν obs is: δf ν (z) = z( Ω) beam ν obs d 2 V(z) dzdω Mmax M min δf dn dm, (5) dm where d 2 V(z)/dzdΩ is the comoving volume per unit redshift per unit solid angle, the solid angle ( Ω) beam = π( θ beam /2) 2, and ν obs / z = ν 0 /(1+z) 2. We calculate the comoving density of halos at different redshifts using the Press-Schechter (PS) approximation for the halo mass function dn/dm. If we define the beam-averaged effective differential antenna temperature δt b using δf ν = 2ν 2 k B δt b ( Ω) beam /c 2, then δt b = c(1+z)4 ν 0 H(z) Mmax M min ν eff δt b,ν0 A dn dm. (6) dm We consider the currently-favored, flat CDM model with cosmological constant ( ΛCDM, Ω 0 = 0.3, λ 0 = 0.7, COBE-normalized, Ω b h 2 = 0.02, h = 0.7), for three values of the primordial power spectrum index n p = 0.9, 1, and 1.1, using the primordial power spectrum of Eisenstein & Hu (1999). Results for δf ν and δt b are plotted in Figure 2. In principle, the variation of δt b with observed frequency implied by the redshift variations in Figure 2 should permit a discrimination between the 21-cm emission from minihalos and the CMB and other backgrounds, due to their very different frequency dependences. However, the average differential brightness temperature of this minihalo background is very low and its evolution is fairly smooth, so such measurement may be difficult in practice with currently planned instruments like LOFAR and SKA. The angular fluctuations in this emission, on the other hand, should be much easier to detect, as discussed in the next section.

6 6 4. Angular Fluctuations in the 21-cm Emission Background The amplitude of q-σ angular fluctuations (i.e. q times the rms value) in the differential antenna temperature is given in the linear regime by δt 2 b 1/2 δt b = qb(z)σ p, (7) where σ p is the rms mass fluctuation at redshift z in a randomly placed cylinder which corresponds to the observational volume defined by the detector angular beam size, θ beam, and frequency bandwidth, ν obs, and b(z) is the bias factor which accounts for the clustering of rare density peaks relative to the mass. We assume b(z) is the flux-weighted average over the mass function of b(m,z) = 1+(νh 2 1)/δ c, the linear bias factor, where ν h = δ c /σ(m), δ c is the value of the linearly extrapolated value of overdensity δρ/ρ corresponding to the epoch when a top-hat collapse reaches infinite density, and σ(m) is the the standard deviation of the density contrast filtered on mass scale M (e.g. Mo & White 1996). For a cylinder of comoving radius R = θ beam (1+z)D A (z)/2, and length L (1+z)cH(z) 1 ( ν/ν) obs, we have : σ 2 p = 8D 2 (z) π 2 R 2 L 2 0 dk 1 0 dx sin2 (klx/2)j 2 1[kR(1 x 2 ) 1/2 ] x 2 (1 x 2 ) (1+fx 2 ) 2P(k) k 2 (8) (Tozzi et al. 2000) (with several typos in the corresponding expression in that paper corrected here), where D(z) δ + (0)/δ + (z) is the linear growth factor, P(k) is the linear power spectrum at z = 0, and the factor (1 + fx 2 ) 2, with f [Ω(z)] 0.6, is the correction to the cylinder length for the departure from Hubble expansion due to peculiar velocities (Kaiser 1987). Illustrative results are plotted for 3-σ fluctuations as a function of θ beam, for z = 7 and 8.5, in Figure 3, along with the expected sensitivity limits for the planned LOFAR (300 m filled aperture) and SKA (1 km filled aperture) arrays. We plot in Figure 4 the predicted spectral variation of these fluctuations vs. redshift z for illustrative beam sizes of θ beam = 9 and 25. These 3-σ fluctuations should be observable with both LOFAR and SKA with integration times of between 100 and 1000 hours. For a 25 beam, for example, 3-σ fluctuations can be detected for untilted ΛCDM by both with a 100 h integration for z and a 1000 h integration for z 11.5, while for a 9 beam, SKA can detect them after 100 h for z 9 and after 1000 h for z 13. Results for different values of z and θ beam are available upon request.

7 7 Acknowledgments Wearegrateful toe. Scannapieco forclarifying theeffects ofbiasandrefereep. Tozzi for his thoughtful comments. This work was supported by European Community RTN contract HPRN-CT RG29185 and grants NASA ATP NAG and NAG and Texas Advanced Research Program REFERENCES Bagla, J. S., Nath, B., & Padmanabhan, T. 1997, MNRAS, 289, 671 Field, G. B. 1958, Proc. I.R.E., 46, 240 Field, G. B. 1959, ApJ, 129, 536 Eisenstein, D. J., & Hu, W. 1999, ApJ, 511, 5 Hogan, C. J., & Rees, M. J. 1979, MNRAS, 188, 791 Iliev, I. T., & Shapiro, P. R. 2001, MNRAS, 325, 468 Iliev, I.T., & Shapiro, P. R. 2002, in TheMass ofgalaxiesat LowandHighRedshift (ESO Astrophysics Symposia), eds. R. Bender & A. Renzini (Heidelberg: Springer-Verlag), in press (astro-ph/ ) Kaiser, N. 1987, MNRAS, 227, 1 Kumar, A., Padmanabhan, T., & Subramanian, K. 1995, MNRAS, 272, 544 Madau, P., Meiksen, A., & Rees, M. J. 1997, ApJ, 475, 429 Mo, H., & White, S. D. M. 1996, MNRAS, 282, 347 Purcell, E. M., & Field, G. B. 1956, ApJ, 124, 542 Rees, M. J. 1999, Physics Reports, 333, 203 Scott, D., & Rees, M. J. 1990, MNRAS, 247, 510 Shapiro, P. R. 2001, in Proceedings of the 20th Texas Symposium on Relativistic Astrophysics and Cosmology, eds. H. Martel and J. C. Wheeler, (AIP Conference Series), pp

8 8 Shapiro, P. R., Iliev, I. T., & Raga, A. C. 1999, MNRAS, 307, 203 Shaver, P. A., Windhorst, R. A., Madau, P., & de Bruyn, A. G. 1999, A&A, 345, 390 Subramanian, K., & Padmanabhan, T. 1993, MNRAS, 265, 101 Tozzi, P., Madau, P., Meiksen, A., & Rees, M. J. 2000, ApJ, 528, 597 Fig. 1. Individual minihalo sources of redshifted 21-cm emission in ΛCDM, redshifts 1+z = 7 (short-dashed line), 10 (solid line), 15 (long-dashed line), and 20 (dotted line) vs. total massofminihalom. Fromtoptobottom: opticaldepthτ ν0 (r = 0)atline-centered frequency ν 0 thru minihalo center, differential antenna temperature δt b, line-integrated differential flux δf(m,z) relative to the CMB, total differential flux per unit frequency F ν0, angular size of minihalo ( θ) halo, and redshifted effective width ν eff (z) of the 21-cm line as observed at z = 0 at received frequency ν rec = ν 0 (1+z) 1. Fig. 2. Minihalo radiation background. Average observed differential antenna temperature δt b and average differential flux per unit frequency δf ν for beam size of θ beam = 10 at the redshifted 21-cm line frequency due to minihalos vs. redshift z for ΛCDM models with power-spectrum tilts n p = 0.9, 1.0, and 1.1, as labelled. Fig. 3. Predicted 3-σ differential antenna temperature fluctuations at z = 7 ((ν rec = MHz; top panel) and z = 8.5 (ν rec = 150 MHz; bottom panel) for bandwidth ν obs = 1MHz vs. angular scale θ beam for ΛCDM models with tilt n p = 0.9, 1.0, and 1.1, as labelled (solid curves). Also indicated is the predicted sensitivity of LOFAR and SKA for a confidence level of 5 times the noise level after integration times of 100 h (dashed lines) and 1000 h (dotted lines), with compact subaperture (horizontal lines) and extended configuration needed to achieve higher resolution (diagonal lines) (see Fig. 4. Predicted 3-σ differential antenna temperature fluctuations at θ beam = 9 (top panel) and 25 (bottom panel) vs. redshift z for ΛCDM models with tilt n p = 0.9, 1.0, and 1.1, as labelled (solid curves). As in Figure 3, we also plot the predicted sensitivity for integration times 100 h (dashed) and 1000 h (dotted) of both LOFAR ( L ) and SKA ( S ), as labelled (for bottom panel, sensitivity curves for LOFAR and SKA are identical), for compact subaperture and assuming rms sensitivity ν 2.4 (see This preprint was prepared with the AAS L A TEX macros v5.0.

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